I am trying to find a closed-form expression for the following sum: $$ \sum_{n=0}^\infty (-1)^n \exp(x)^n \exp(y)^{n^2} \tag{1} $$ This sum seems to be very related to the Jacobi theta function: $$ \vartheta_4(z, \tau) = \sum_{n=-\infty}^\infty (-1)^n \eta^n q^{n^2} $$ where $q = \exp(\pi i \tau)$ is the nome and $\eta = \exp(2 \pi i z)$.
However, first, it might be trivial but the range of the sums is different and I haven't been able to express $\vartheta_4(z, \tau)$ with a sum ranging from $0$ to $\infty$. Second, it seems that the Jacobi theta function is defined with $|q| < 1$ which might not be the case in my situation, is condition indispensable? when I compute (1) with $\mathrm{mpmath.nsum}$ with $|q| > 1$, it seems to work.
Any hints and suggestions would be appreciated. Thank you!